Ramsey Graphs Induce Subgraphs of Many Different Sizes

Abstract

Abstract. A graph on n vertices is said to be C-Ramsey if every clique or independent set of the graph has size at most C log n. The only known constructions of Ramsey graphs are probabilistic in nature, and it is generally believed that such graphs possess many of the same properties as dense random graphs. Here, we demonstrate one such property: for any fixed C > 0, every C-Ramsey graph on n vertices induces subgraphs of at least n 2−o(1) distinct sizes. This near-optimal result is closely related to two unresolved conjectures, the first due to Erdős and McKay and the second due to Erdős, Faudree and Sós, both from 1992.

Proof of a conjecture on induced subgraphs of Ramsey graphs

“…In fact, recent developments bounding |Φ(G)| due to Narayanan, Sahasrabudhe and Tomon [30], and ourselves [27], make this idea seem even more promising. In [30], the authors made the simple observation (using the pigeonhole principle) that in any n-vertex graph G, there is a set A of √ n vertices with degrees lying in an interval of length √ n. If G is diverse, and U is a random vertex set of linear size, then the degrees d U (x), for x ∈ A, are likely to take n 1/2−o(1) different values, very tightly packed in an interval of length O( √ n). By augmenting U with different combinations of vertices in A, we can obtain subgraphs of many different sizes, all lying in a fixed interval of length O(n).…”

“…We will prove Lemma 4.3 in Section 4.2, using some ideas from [27,30]. To interpret its conclusion in words, it says that one can obtain Ω n Z √ n D induced subgraphs with different numbers of edges, by augmenting W ∪ U with different subsets Z ⊆ M of size n Z .…”

Ramsey Graphs Induce Subgraphs of Quadratically Many Sizes

“…First, we can repeat the above argument for many different values of p, and second, instead of deleting single vertices, we might hope to obtain a richer variety of subgraphs by adding and deleting different combinations of vertices. Narayanan, Sahasrabudhe and Tomon [21] combined both these ideas, as follows.…”

“…Unfortunately, while by construction the degrees d U (w), for w ∈ W , are different, it does not follow that the d U ∪Z (w) are also different. In order to make this approach work, the authors of [21] came up with a way to introduce some limited randomness into the choice of the sets Z, and with a rather delicate combination of concentration and anticoncentration arguments they were able to show that there are likely to be many different values of d U ∪Z (w).…”

Anticoncentration for subgraph statistics

“…It is also interesting to study the probabilities Pr(X G,k = ℓ) for restricted classes of (hyper)graphs G. If these probabilities are small it would seem to give some evidence that the graphs in question are very "diverse" or "disordered". In particular, say that a graph is C-Ramsey if it has no clique or independent set of size C log 2 n. There has been a lot of work on diversity of Ramsey graphs from various points of view; in particular, Kwan and Sudakov [28] recently resolved a conjecture of Erdős, Faudree and Sós which effectively says that if G is an O(1)-Ramsey graph then for many values of k, the random variables X G,k have large support (see also [2,6,5,1,34,29] for related work). It would be very interesting to study the probabilities Pr(X G,k = ℓ) for Ramsey graphs.…”